Decompactification Limits of Non-Compact Gauge Theory

1. Problem Statement

The paper addresses a central tension in the Swampland program of quantum gravity: the conjecture that consistent theories of quantum gravity (the "Landscape") cannot possess global symmetries, including generalized global symmetries.

• The Conflict: Non-compact gauge theories (specifically those with gauge group $G = \mathbb{R}$) appear to inevitably possess unbreakable generalized global symmetries.
  • 1-form Symmetry: In $\mathbb{R}$ gauge theory, Wilson lines $W_q$ carry continuous charges $q \in \mathbb{R}$. Unlike compact $U(1)$ theories where charges are quantized ($q \in \mathbb{Z}$), breaking the 1-form symmetry requires screening all possible charges. A finite or countable set of matter fields cannot screen the uncountable continuum of $\mathbb{R}$ charges.
  • 0-form Symmetry: Arguments suggest that if one adds matter fields with charges like $1$ and $\sqrt{2}$, a global symmetry emerges where these fields transform differently, violating the "No Global Symmetries" conjecture.
• The Consequence: Standard arguments imply that $\mathbb{R}$ gauge theories belong to the Swampland (inconsistent with quantum gravity) because their associated global symmetries cannot be broken by a finite number of fields without violating the Covariant Entropy Bound (CEB) due to the infinite species count.

2. Methodology

The authors investigate a loophole: Can adding an uncountable infinity of fields break these symmetries, and if so, what is the physical interpretation of such a theory?

The methodology proceeds in two main stages:

A. Breaking Symmetries with Uncountably Many Fields

1. Field Content: The authors propose adding a scalar field $\phi_q$ for every charge $q \in \mathbb{R}$.
2. Breaking 1-form Symmetry: With a field for every $q$, every Wilson line $W_q$ can end on a local operator $\phi_q$, explicitly breaking the 1-form symmetry.
3. Breaking 0-form Symmetry:
   • They analyze the constraints on global charges $Q_q$ imposed by gauge-invariant interactions (e.g., $\phi_{q_1}\phi_{q_2}\phi_{-(q_1+q_2)}$).
   • They show that while algebraic constraints allow for a global symmetry (where $Q_q$ is a linear function of $q$), the Noether current $J_\mu = \int dq \, Q_q J^q_\mu$ becomes ill-defined.
   • Using Riemann integrability arguments, they prove that for the topological operator generating the symmetry to be well-defined, $Q_q$ must be of the form $kq$. If $k \neq 0$, the "global" symmetry is actually a gauge transformation (redundant). If $k=0$, the symmetry is trivial.
   • Conclusion: The topological operator cannot be defined for any non-trivial global symmetry; thus, the symmetry is broken in the modern sense.

B. Decompactification to Higher Dimensions

Recognizing that an Effective Field Theory (EFT) with uncountably many fields in $d$ dimensions is pathological (the species scale $\Lambda_{sp} \to 0$), the authors propose that this theory is actually a decompactification limit of a $(d+1)$-dimensional theory.

1. Kaluza-Klein (KK) on $\mathbb{R}$: They treat the continuous charge $q$ as the momentum in an extra non-compact dimension $y \in \mathbb{R}$.
2. Fourier Transform: The infinite set of fields $\phi_q(x)$ is re-packaged via Fourier transform into a single scalar field $\Phi(x, y)$ in $D = d+1$ dimensions.
3. Metric Ansatz: They utilize a KK reduction ansatz where the $D$-dimensional metric $G_{MN}$ includes the $d$-dimensional metric $g_{\mu\nu}$, the gauge field $A_\mu$, and a modulus $\lambda$ (dilaton).
4. Constraints: They derive specific constraints on the $d$-dimensional parameters required for this decompactification to hold:
   • Mass spectrum: $m_q^2 = M^2 + \lambda q^2$.
   • Coupling constraints: The gauge coupling $g$ and modulus $\lambda$ must satisfy $M_{Pl;d}^{d-2} g^2 = \lambda$.
   • Interaction constraints: Cubic couplings must be independent of charge ($c_{q,q'} = c_\Phi$).

3. Key Contributions and Results

• Resolution of the Global Symmetry Paradox: The paper demonstrates that $\mathbb{R}$ gauge theories can be consistent with the "No Global Symmetries" conjecture if one accepts an uncountable spectrum of fields. The global symmetries are broken because the topological operators generating them become mathematically ill-defined (non-integrable) in the continuum limit.
• Decompactification Mechanism: The authors show that the "pathological" $d$-dimensional theory with a continuum of fields is physically equivalent to a well-behaved $(d+1)$-dimensional theory with a single scalar field coupled to gravity.
  • The $\mathbb{R}$ gauge symmetry in $d$ dimensions corresponds to diffeomorphisms (shifts) in the extra non-compact dimension $y$ in $D$ dimensions.
  • This effectively "undoes" a Kaluza-Klein reduction on a non-compact space.
• Weak Gravity Conjecture (WGC) Consistency:
  • The authors analyze the WGC for $\mathbb{R}$ gauge theory.
  • They find that for the theory to decompactify consistently, the $D$-dimensional scalar mass $M$ must be zero ($M=0$).
  • This implies the $d$-dimensional particles satisfy $m_q^2 = \lambda q^2$, meaning they are extremal (saturating the WGC bound).
  • They argue that for $\mathbb{R}$, the WGC requires that for any non-zero charge $Q$, there exists a particle with charge $\le Q$ satisfying the bound. The continuum of extremal particles satisfies this.
• Vanishing Gauge Coupling and Species Scale:
  • In the decompactification limit (volume of $\mathbb{R} \to \infty$), the $d$-dimensional Planck mass $M_{Pl;d}$ diverges, causing the $d$-dimensional Newton constant $G_N^{(d)}$ to vanish.
  • Consequently, the gauge coupling $g^2 \propto G_N^{(d)}$ vanishes.
  • The pure gauge theory becomes a topological BF theory ($S \sim \int B \wedge F$) with no propagating degrees of freedom, consistent with the fact that the gauge symmetry is now a diffeomorphism in the higher dimension.
  • Crucially, while the $d$-dimensional species scale vanishes (making the EFT invalid), the $D$-dimensional species scale is finite, restoring the validity of the theory as a higher-dimensional EFT.

4. Significance

• Reframing Non-Compact Gauge Theories: The paper provides a concrete mechanism by which non-compact gauge theories, previously thought to be strictly in the Swampland, can exist as limits of consistent quantum gravity theories. It shifts the perspective from "bad $d$-dimensional EFT" to "good $D$-dimensional EFT."
• Role of Uncountable Spectra: It challenges the assumption that infinite species always lead to inconsistency. It shows that if the infinite spectrum arises from a geometric decompactification (momentum modes in a non-compact direction), the theory remains consistent.
• Swampland Constraints: The work refines the application of the Weak Gravity Conjecture and the No Global Symmetries conjecture to non-compact groups, suggesting that the "loophole" of adding infinite fields is not just a mathematical trick but a signal of a higher-dimensional origin.
• Future Directions: The authors suggest this framework could be generalized to other non-compact groups (using Haar measures and Plancherel's theorem) and invites phenomenological investigation into scenarios where such continuum spectra might arise naturally.

In summary, the paper resolves the apparent inconsistency of $\mathbb{R}$ gauge theories by demonstrating that they are effectively Kaluza-Klein reductions on non-compact spaces, where the "global symmetries" are artifacts of the lower-dimensional truncation and the "infinite species" are simply the momentum modes of a single higher-dimensional field.